Does NEAT require only connection genes to be marked with a global innovation number?
It is actually the other way around: connection IDs is what is debated!

Nodes always have innovation IDs (in the image, it is just their identifying number).

Node IDs are sufficient to identify connections. If a connection links nodes 3 and 6, then it is the same as another connection linking nodes 3 and 6: no need for an extra ID. So why the extra innovation IDs then?

On the one hand, this is an implementation choice: maybe these extra IDs would allow you to create a more complex but faster code?

On the other hand, there is a debate around whether a connection between two nodes means the same thing at different times in evolution. If you have no innovation IDs, then you cannot tell apart an old connection from 3 and 6 and another that was independently created later in a different genome (imagine the old connection was removed first). Is this relevant? As said, it is an open debate. Surely, it is not crucial at a basic level!

This question (and my answers) is related to this other question on Stack Overflow.